Real-variable Characterizations of Orlicz-Hardy Spaces on Strongly Lipschitz Domains of $\mathbb{R}^n$

Let $Ω$ be a strongly Lipschitz domain of $\mathbb{R}^n$, whose complement in $\mathbb{R}^n$ is unbounded. Let $L$ be a second order divergence form elliptic operator on $L^2 (Ω)$ with the Dirichlet boundary condition, and the heat semigroup generated by $L$ have the Gaussian property $(G_{\mathrm{diam}(Ω)})$ with the regularity of their kernels measured by $μ\in(0,1]$, where $\mathrm{diam}(Ω)$ denotes the diameter of $Ω$. Let $Φ$ be a continuous, strictly increasing, subadditive and positive function on $(0,\infty)$ of upper type 1 and of strictly critical lower type $p_Φ\in(n/(n+μ),1]$. In this paper, the authors introduce the Orlicz-Hardy space $H_{Φ,\,r}(Ω)$ by restricting arbitrary elements of the Orlicz-Hardy space $H_Φ(\mathbb{R}^n)$ to $\boz$ and establish its atomic decomposition by means of the Lusin area function associated with $\{e^{-tL}\}_{t\ge0}$. Applying this, the authors obtain two equivalent characterizations of $H_{Φ,\,r}(\boz)$ in terms of the nontangential maximal function and the Lusin area function associated with the heat semigroup generated by $L$.

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